Abstract

A closed-form analytical expression is obtained for the spatio-temporal correlation function of the scattered radiation detected in fiber-based optical coherence tomography (OCT), assuming a clean optical system arrangement in the OCT sample arm. It is shown that the transverse flow component causes purely translational speckle motion with the predicted speckle velocity 2x higher than the velocity of the flowing particles as would be observed in the image plane under incoherent illumination. It is also shown that both speckle velocity and speckle radius do not depend on the position of the scattering volume relative to the focal plane, hence the derived correlation function is independent of the position of the scattering volume relative to the focal plane. Although the analytical results are obtained for a clean optical system arrangement, they can be used with high accuracy in most practical implementations of fiber based OCT. Validation experiments in control scattering phantoms with varying liquid viscosities show excellent agreement with the developed theoretical model, under both no-flow and flow conditions. Accurate viscosity determinations enabled by this methodology may have applications to non-invasive glucose measurements in medicine.

© 2017 Optical Society of America

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References

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    [Crossref] [PubMed]
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  4. J. Lee, W. Wu, F. Lesage, and D. A. Boas, “Multiple-capillary measurement of RBC speed, flux, and density with optical coherence tomography,” J. Cereb. Blood Flow Metab. 33(11), 1707–1710 (2013).
    [Crossref] [PubMed]
  5. V. Kalchenko, K. Ziv, Y. Addadi, N. Madar-Balakirski, I. Meglinski, M. Neeman, and A. Harmelin, “Combined application of dynamic light scattering imaging and fluorescence intravital microscopy in vascular biology,” Laser Phys. Lett. 7(8), 603–606 (2010).
    [Crossref]
  6. A. Mariampillai, B. A. Standish, E. H. Moriyama, M. Khurana, N. R. Munce, M. K. K. Leung, J. Jiang, A. Cable, B. C. Wilson, I. A. Vitkin, and V. X. D. Yang, “Speckle variance detection of microvasculature using swept-source optical coherence tomography,” Opt. Lett. 33(13), 1530–1532 (2008).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
  13. I. Popov and A. Vitkin, “Dynamic light scattering by flowing Brownian particles measured with optical coherence tomography: impact of the optical system,” J. Biomed. Opt. 21(1), 017002 (2016).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  23. L. M. Veselov and I. A. Popov, “Characteristics of scattered radiation under coherent-beam scanning along a rough surface,” Opt. Spectrosc. 70, 636–639 (1991).
  24. N. M. Temme, “Chapter 7: Error Functions, Dawson’s and Fresnel Integrals,” in NIST Handbook of Mathematical Functions, F. W. J. Olver, ed., (Cambridge University, 2010).
  25. B. Davoudi, A. Lindenmaier, B. A. Standish, G. Allo, K. Bizheva, and A. Vitkin, “Noninvasive in vivo structural and vascular imaging of human oral tissues with spectral domain optical coherence tomography,” Biomed. Opt. Express 3(5), 826–839 (2012).
    [Crossref] [PubMed]
  26. C. R. Hammond, CRC Handbook of Chemistry and Physics (2016).
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    [Crossref] [PubMed]
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    [Crossref]

2016 (1)

I. Popov and A. Vitkin, “Dynamic light scattering by flowing Brownian particles measured with optical coherence tomography: impact of the optical system,” J. Biomed. Opt. 21(1), 017002 (2016).
[Crossref] [PubMed]

2015 (1)

2014 (1)

I. Popov, A. S. Weatherbee, and I. A. Vitkin, “Dynamic light scattering arising from flowing Brownian particles: analytical model in optical coherence tomography conditions,” J. Biomed. Opt. 19(12), 127004 (2014).
[Crossref] [PubMed]

2013 (2)

J. Lee, W. Wu, F. Lesage, and D. A. Boas, “Multiple-capillary measurement of RBC speed, flux, and density with optical coherence tomography,” J. Cereb. Blood Flow Metab. 33(11), 1707–1710 (2013).
[Crossref] [PubMed]

N. Weiss, T. G. van Leeuwen, and J. Kalkman, “Localized measurement of longitudinal and transverse flow velocities in colloidal suspensions using optical coherence tomography,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 88(4), 042312 (2013).
[Crossref] [PubMed]

2012 (2)

2011 (1)

H. Ullah, A. Mariampillai, M. Ikram, and I. A. Vitkin, “Can temporal analysis of optical coherence tomography statistics report on dextrorotatory glucose levels in blood?” Laser Phys. 21(11), 1962–1971 (2011).
[Crossref]

2010 (1)

V. Kalchenko, K. Ziv, Y. Addadi, N. Madar-Balakirski, I. Meglinski, M. Neeman, and A. Harmelin, “Combined application of dynamic light scattering imaging and fluorescence intravital microscopy in vascular biology,” Laser Phys. Lett. 7(8), 603–606 (2010).
[Crossref]

2008 (1)

2006 (1)

J. Rogowska, N. Patel, S. Plummer, and M. E. Brezinski, “Quantitative optical coherence tomographic elastography: method for assessing arterial mechanical properties,” Br. J. Radiol. 79(945), 707–711 (2006).
[Crossref] [PubMed]

1998 (2)

1991 (1)

L. M. Veselov and I. A. Popov, “Characteristics of scattered radiation under coherent-beam scanning along a rough surface,” Opt. Spectrosc. 70, 636–639 (1991).

1988 (1)

D. A. Berstad, B. Knapstad, M. Lamvik, P. A. Skjølsvik, K. Tørklep, and H. A. Øye, “Accurate measurements of the viscosity of water in the temperature range 19.5–25.5 C,” Physica A 151(2), 246–280 (1988).
[Crossref]

1986 (3)

1985 (1)

1976 (1)

J. C. Dainty, “The statistics of speckle patterns,” Prog. Opt. 14, 3–46 (1976).

1970 (1)

Addadi, Y.

V. Kalchenko, K. Ziv, Y. Addadi, N. Madar-Balakirski, I. Meglinski, M. Neeman, and A. Harmelin, “Combined application of dynamic light scattering imaging and fluorescence intravital microscopy in vascular biology,” Laser Phys. Lett. 7(8), 603–606 (2010).
[Crossref]

Aizu, Y.

Allo, G.

Asakura, T.

Berstad, D. A.

D. A. Berstad, B. Knapstad, M. Lamvik, P. A. Skjølsvik, K. Tørklep, and H. A. Øye, “Accurate measurements of the viscosity of water in the temperature range 19.5–25.5 C,” Physica A 151(2), 246–280 (1988).
[Crossref]

Bizheva, K.

Boas, D. A.

J. Lee, W. Wu, F. Lesage, and D. A. Boas, “Multiple-capillary measurement of RBC speed, flux, and density with optical coherence tomography,” J. Cereb. Blood Flow Metab. 33(11), 1707–1710 (2013).
[Crossref] [PubMed]

J. Lee, W. Wu, J. Y. Jiang, B. Zhu, and D. A. Boas, “Dynamic light scattering optical coherence tomography,” Opt. Express 20(20), 22262–22277 (2012).
[Crossref] [PubMed]

Brezinski, M. E.

J. Rogowska, N. Patel, S. Plummer, and M. E. Brezinski, “Quantitative optical coherence tomographic elastography: method for assessing arterial mechanical properties,” Br. J. Radiol. 79(945), 707–711 (2006).
[Crossref] [PubMed]

Cable, A.

Collins, S. A.

Dainty, J. C.

J. C. Dainty, “The statistics of speckle patterns,” Prog. Opt. 14, 3–46 (1976).

Davoudi, B.

Hanson, S. G.

Harmelin, A.

V. Kalchenko, K. Ziv, Y. Addadi, N. Madar-Balakirski, I. Meglinski, M. Neeman, and A. Harmelin, “Combined application of dynamic light scattering imaging and fluorescence intravital microscopy in vascular biology,” Laser Phys. Lett. 7(8), 603–606 (2010).
[Crossref]

Ikram, M.

H. Ullah, A. Mariampillai, M. Ikram, and I. A. Vitkin, “Can temporal analysis of optical coherence tomography statistics report on dextrorotatory glucose levels in blood?” Laser Phys. 21(11), 1962–1971 (2011).
[Crossref]

Jiang, J.

Jiang, J. Y.

Kalchenko, V.

V. Kalchenko, K. Ziv, Y. Addadi, N. Madar-Balakirski, I. Meglinski, M. Neeman, and A. Harmelin, “Combined application of dynamic light scattering imaging and fluorescence intravital microscopy in vascular biology,” Laser Phys. Lett. 7(8), 603–606 (2010).
[Crossref]

Kalkman, J.

N. Weiss, T. G. van Leeuwen, and J. Kalkman, “Simultaneous measurement of localized diffusion and flow using optical coherence tomography,” Opt. Express 23(3), 3448–3459 (2015).
[Crossref] [PubMed]

N. Weiss, T. G. van Leeuwen, and J. Kalkman, “Localized measurement of longitudinal and transverse flow velocities in colloidal suspensions using optical coherence tomography,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 88(4), 042312 (2013).
[Crossref] [PubMed]

Khurana, M.

Knapstad, B.

D. A. Berstad, B. Knapstad, M. Lamvik, P. A. Skjølsvik, K. Tørklep, and H. A. Øye, “Accurate measurements of the viscosity of water in the temperature range 19.5–25.5 C,” Physica A 151(2), 246–280 (1988).
[Crossref]

Lamvik, M.

D. A. Berstad, B. Knapstad, M. Lamvik, P. A. Skjølsvik, K. Tørklep, and H. A. Øye, “Accurate measurements of the viscosity of water in the temperature range 19.5–25.5 C,” Physica A 151(2), 246–280 (1988).
[Crossref]

Lee, J.

J. Lee, W. Wu, F. Lesage, and D. A. Boas, “Multiple-capillary measurement of RBC speed, flux, and density with optical coherence tomography,” J. Cereb. Blood Flow Metab. 33(11), 1707–1710 (2013).
[Crossref] [PubMed]

J. Lee, W. Wu, J. Y. Jiang, B. Zhu, and D. A. Boas, “Dynamic light scattering optical coherence tomography,” Opt. Express 20(20), 22262–22277 (2012).
[Crossref] [PubMed]

Lesage, F.

J. Lee, W. Wu, F. Lesage, and D. A. Boas, “Multiple-capillary measurement of RBC speed, flux, and density with optical coherence tomography,” J. Cereb. Blood Flow Metab. 33(11), 1707–1710 (2013).
[Crossref] [PubMed]

Leung, M. K. K.

Lindenmaier, A.

Madar-Balakirski, N.

V. Kalchenko, K. Ziv, Y. Addadi, N. Madar-Balakirski, I. Meglinski, M. Neeman, and A. Harmelin, “Combined application of dynamic light scattering imaging and fluorescence intravital microscopy in vascular biology,” Laser Phys. Lett. 7(8), 603–606 (2010).
[Crossref]

Mariampillai, A.

Meglinski, I.

V. Kalchenko, K. Ziv, Y. Addadi, N. Madar-Balakirski, I. Meglinski, M. Neeman, and A. Harmelin, “Combined application of dynamic light scattering imaging and fluorescence intravital microscopy in vascular biology,” Laser Phys. Lett. 7(8), 603–606 (2010).
[Crossref]

Moriyama, E. H.

Munce, N. R.

Neeman, M.

V. Kalchenko, K. Ziv, Y. Addadi, N. Madar-Balakirski, I. Meglinski, M. Neeman, and A. Harmelin, “Combined application of dynamic light scattering imaging and fluorescence intravital microscopy in vascular biology,” Laser Phys. Lett. 7(8), 603–606 (2010).
[Crossref]

Øye, H. A.

D. A. Berstad, B. Knapstad, M. Lamvik, P. A. Skjølsvik, K. Tørklep, and H. A. Øye, “Accurate measurements of the viscosity of water in the temperature range 19.5–25.5 C,” Physica A 151(2), 246–280 (1988).
[Crossref]

Patel, N.

J. Rogowska, N. Patel, S. Plummer, and M. E. Brezinski, “Quantitative optical coherence tomographic elastography: method for assessing arterial mechanical properties,” Br. J. Radiol. 79(945), 707–711 (2006).
[Crossref] [PubMed]

Plummer, S.

J. Rogowska, N. Patel, S. Plummer, and M. E. Brezinski, “Quantitative optical coherence tomographic elastography: method for assessing arterial mechanical properties,” Br. J. Radiol. 79(945), 707–711 (2006).
[Crossref] [PubMed]

Popov, I.

I. Popov and A. Vitkin, “Dynamic light scattering by flowing Brownian particles measured with optical coherence tomography: impact of the optical system,” J. Biomed. Opt. 21(1), 017002 (2016).
[Crossref] [PubMed]

I. Popov, A. S. Weatherbee, and I. A. Vitkin, “Dynamic light scattering arising from flowing Brownian particles: analytical model in optical coherence tomography conditions,” J. Biomed. Opt. 19(12), 127004 (2014).
[Crossref] [PubMed]

Popov, I. A.

L. M. Veselov and I. A. Popov, “Characteristics of scattered radiation under coherent-beam scanning along a rough surface,” Opt. Spectrosc. 70, 636–639 (1991).

Rogowska, J.

J. Rogowska, N. Patel, S. Plummer, and M. E. Brezinski, “Quantitative optical coherence tomographic elastography: method for assessing arterial mechanical properties,” Br. J. Radiol. 79(945), 707–711 (2006).
[Crossref] [PubMed]

Rose, B.

Schmitt, J.

Skjølsvik, P. A.

D. A. Berstad, B. Knapstad, M. Lamvik, P. A. Skjølsvik, K. Tørklep, and H. A. Øye, “Accurate measurements of the viscosity of water in the temperature range 19.5–25.5 C,” Physica A 151(2), 246–280 (1988).
[Crossref]

Sorensen, C. M.

Standish, B. A.

Taylor, T. W.

Tørklep, K.

D. A. Berstad, B. Knapstad, M. Lamvik, P. A. Skjølsvik, K. Tørklep, and H. A. Øye, “Accurate measurements of the viscosity of water in the temperature range 19.5–25.5 C,” Physica A 151(2), 246–280 (1988).
[Crossref]

Ullah, H.

H. Ullah, A. Mariampillai, M. Ikram, and I. A. Vitkin, “Can temporal analysis of optical coherence tomography statistics report on dextrorotatory glucose levels in blood?” Laser Phys. 21(11), 1962–1971 (2011).
[Crossref]

Ushizaka, T.

van Leeuwen, T. G.

N. Weiss, T. G. van Leeuwen, and J. Kalkman, “Simultaneous measurement of localized diffusion and flow using optical coherence tomography,” Opt. Express 23(3), 3448–3459 (2015).
[Crossref] [PubMed]

N. Weiss, T. G. van Leeuwen, and J. Kalkman, “Localized measurement of longitudinal and transverse flow velocities in colloidal suspensions using optical coherence tomography,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 88(4), 042312 (2013).
[Crossref] [PubMed]

Veselov, L. M.

L. M. Veselov and I. A. Popov, “Characteristics of scattered radiation under coherent-beam scanning along a rough surface,” Opt. Spectrosc. 70, 636–639 (1991).

Vitkin, A.

I. Popov and A. Vitkin, “Dynamic light scattering by flowing Brownian particles measured with optical coherence tomography: impact of the optical system,” J. Biomed. Opt. 21(1), 017002 (2016).
[Crossref] [PubMed]

B. Davoudi, A. Lindenmaier, B. A. Standish, G. Allo, K. Bizheva, and A. Vitkin, “Noninvasive in vivo structural and vascular imaging of human oral tissues with spectral domain optical coherence tomography,” Biomed. Opt. Express 3(5), 826–839 (2012).
[Crossref] [PubMed]

Vitkin, I. A.

I. Popov, A. S. Weatherbee, and I. A. Vitkin, “Dynamic light scattering arising from flowing Brownian particles: analytical model in optical coherence tomography conditions,” J. Biomed. Opt. 19(12), 127004 (2014).
[Crossref] [PubMed]

H. Ullah, A. Mariampillai, M. Ikram, and I. A. Vitkin, “Can temporal analysis of optical coherence tomography statistics report on dextrorotatory glucose levels in blood?” Laser Phys. 21(11), 1962–1971 (2011).
[Crossref]

A. Mariampillai, B. A. Standish, E. H. Moriyama, M. Khurana, N. R. Munce, M. K. K. Leung, J. Jiang, A. Cable, B. C. Wilson, I. A. Vitkin, and V. X. D. Yang, “Speckle variance detection of microvasculature using swept-source optical coherence tomography,” Opt. Lett. 33(13), 1530–1532 (2008).
[Crossref] [PubMed]

Weatherbee, A. S.

I. Popov, A. S. Weatherbee, and I. A. Vitkin, “Dynamic light scattering arising from flowing Brownian particles: analytical model in optical coherence tomography conditions,” J. Biomed. Opt. 19(12), 127004 (2014).
[Crossref] [PubMed]

Weiss, N.

N. Weiss, T. G. van Leeuwen, and J. Kalkman, “Simultaneous measurement of localized diffusion and flow using optical coherence tomography,” Opt. Express 23(3), 3448–3459 (2015).
[Crossref] [PubMed]

N. Weiss, T. G. van Leeuwen, and J. Kalkman, “Localized measurement of longitudinal and transverse flow velocities in colloidal suspensions using optical coherence tomography,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 88(4), 042312 (2013).
[Crossref] [PubMed]

Wilson, B. C.

Wu, W.

J. Lee, W. Wu, F. Lesage, and D. A. Boas, “Multiple-capillary measurement of RBC speed, flux, and density with optical coherence tomography,” J. Cereb. Blood Flow Metab. 33(11), 1707–1710 (2013).
[Crossref] [PubMed]

J. Lee, W. Wu, J. Y. Jiang, B. Zhu, and D. A. Boas, “Dynamic light scattering optical coherence tomography,” Opt. Express 20(20), 22262–22277 (2012).
[Crossref] [PubMed]

Yang, V. X. D.

Yoshimura, T.

Yura, H. T.

Zhu, B.

Ziv, K.

V. Kalchenko, K. Ziv, Y. Addadi, N. Madar-Balakirski, I. Meglinski, M. Neeman, and A. Harmelin, “Combined application of dynamic light scattering imaging and fluorescence intravital microscopy in vascular biology,” Laser Phys. Lett. 7(8), 603–606 (2010).
[Crossref]

Appl. Opt. (3)

Biomed. Opt. Express (1)

Br. J. Radiol. (1)

J. Rogowska, N. Patel, S. Plummer, and M. E. Brezinski, “Quantitative optical coherence tomographic elastography: method for assessing arterial mechanical properties,” Br. J. Radiol. 79(945), 707–711 (2006).
[Crossref] [PubMed]

J. Biomed. Opt. (2)

I. Popov, A. S. Weatherbee, and I. A. Vitkin, “Dynamic light scattering arising from flowing Brownian particles: analytical model in optical coherence tomography conditions,” J. Biomed. Opt. 19(12), 127004 (2014).
[Crossref] [PubMed]

I. Popov and A. Vitkin, “Dynamic light scattering by flowing Brownian particles measured with optical coherence tomography: impact of the optical system,” J. Biomed. Opt. 21(1), 017002 (2016).
[Crossref] [PubMed]

J. Cereb. Blood Flow Metab. (1)

J. Lee, W. Wu, F. Lesage, and D. A. Boas, “Multiple-capillary measurement of RBC speed, flux, and density with optical coherence tomography,” J. Cereb. Blood Flow Metab. 33(11), 1707–1710 (2013).
[Crossref] [PubMed]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Laser Phys. (1)

H. Ullah, A. Mariampillai, M. Ikram, and I. A. Vitkin, “Can temporal analysis of optical coherence tomography statistics report on dextrorotatory glucose levels in blood?” Laser Phys. 21(11), 1962–1971 (2011).
[Crossref]

Laser Phys. Lett. (1)

V. Kalchenko, K. Ziv, Y. Addadi, N. Madar-Balakirski, I. Meglinski, M. Neeman, and A. Harmelin, “Combined application of dynamic light scattering imaging and fluorescence intravital microscopy in vascular biology,” Laser Phys. Lett. 7(8), 603–606 (2010).
[Crossref]

Opt. Express (3)

Opt. Lett. (1)

Opt. Spectrosc. (1)

L. M. Veselov and I. A. Popov, “Characteristics of scattered radiation under coherent-beam scanning along a rough surface,” Opt. Spectrosc. 70, 636–639 (1991).

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

N. Weiss, T. G. van Leeuwen, and J. Kalkman, “Localized measurement of longitudinal and transverse flow velocities in colloidal suspensions using optical coherence tomography,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 88(4), 042312 (2013).
[Crossref] [PubMed]

Physica A (1)

D. A. Berstad, B. Knapstad, M. Lamvik, P. A. Skjølsvik, K. Tørklep, and H. A. Øye, “Accurate measurements of the viscosity of water in the temperature range 19.5–25.5 C,” Physica A 151(2), 246–280 (1988).
[Crossref]

Prog. Opt. (1)

J. C. Dainty, “The statistics of speckle patterns,” Prog. Opt. 14, 3–46 (1976).

Other (8)

BIPM, IEC, ISO IFCC, and IUPAP IUPAC. “OIML 1995 Guide to the Expression of Uncertainty in Measurement.” ISO, Geneva 3 (1995).

N. M. Temme, “Chapter 7: Error Functions, Dawson’s and Fresnel Integrals,” in NIST Handbook of Mathematical Functions, F. W. J. Olver, ed., (Cambridge University, 2010).

C. R. Hammond, CRC Handbook of Chemistry and Physics (2016).

V. X. D. Yang and I. A. Vitkin, “Principles of Doppler OCT,” in Optical Coherence Tomography in Cardiovascular Research, E. Regar, T.G. van Leeuwen, P. W. Serruys eds., (Informa Healthcare, London, 2007).

A. Yariv, Quantum Electronics (Wiley, 1989), pp. 676.

Hermite function. M.V. Fedoryuk (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Hermite_function&oldid=18370 Accessed August 24, 2016.

B. J. Berne and R. Pecora, Dynamic Light Scattering (Dover Publications, New York, 2000).

G. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed., 2nd ed., (Springer, 1984), pp. 342.

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Figures (3)

Fig. 1
Fig. 1 (a) OCT set-up. (b) Illumination and observation geometry of scattered radiation in an OCT sample arm. Fc and Fs are the focal distances of the collimating and sample lenses, respectively; z1 is the axial offset between the scattering volume and the beam waist. The collimating and sample lenses and the Gaussian diaphragm (Gaussian variation of transparency in the radial direction) form what is called the ‘clean optical system’. Note that the diaphragm need not be physically present in the set-up, as its role is played by the acceptance angle of the fiber.
Fig. 2
Fig. 2 Power spectra of backscattered radiation from stagnant Brownian particles (polystyrene microspheres) in the aqueous solutions of 0 mM, 800 mM, and 1600 mM glucose. Symbols are experimentally obtained power spectrum values; solid lines are theoretical fits of Eq. (20) with τb as the fitting parameter.
Fig. 3
Fig. 3 Power spectra of backscattered radiation from flowing Brownian particles in solutions containing 0, 800 and 1600 mM glucose. Crosses, triangles and circles are experimentally obtained power spectrum values; solid lines are theoretical dependencies given by Eqs. (18) and (19).

Tables (1)

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Table 1 Correlation time, diffusivity and viscosity in stagnant suspensions of microspheres in water with varying amounts of dissolved glucose. All quoted errors were calculated following [28], and are given as the RMS error. The last italicized row shows the corresponding viscosity results measured under flow conditions (see Fig. 3 and text).

Equations (21)

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E in (r,t)= E in0 w 0 w exp[ ( 1 w 2 ik 2ρ )( x 2 + y 2 )+ikziωt+i φ G ]exp[ ( z z 1 l c /2 ) 2 ].
C s ( X 1 ,Z, X 2 ,Z,τ )= E s ( X 1 ,Z, t 1 ) E s * ( X 2 ,Z, t 2 ) = C sb ( τ ) C st ( X 1 ,Z, X 2 ,Z,τ )
C sb ( τ )=exp( ( 2k ) 2 D d τ )
C st ( X 1 ,Z, X 2 ,Z,τ )= E in ( x,z,0 )K( x,z, X 1 ,Z ) E in * ( x+ v xy τ,z+ v z τ ) K * ( x+ v xy τ,z+ v z τ, X 2 ,Z ) dxdydz
K( r,R )= ik 2πB exp{ ik 2B ( D X 2 2xX+A x 2 )+ik( zZ ) }
A= F c F s ,B= F c z 1 F s 2i F c F s k q 2 ,C=0,D= n F s F c
q  N A f F c    λ 0 F c π w f  =  2 F c k 0 w f  =  2 F s k 0 w 0
K( r,R )= ik 2π( z+i z F ) exp{ ( 1 w 2 + ik 2ρ )[ ( x+MX ) 2 +( n1 ) ( MX ) 2 ]+ik( zZ ) }
w=w(z)= w 0 1+ ( z z F ) 2 , ρ=ρ(z)=z+ z F 2 z .
exp[ 2 ( l c /2 ) 2 ( z z 1 + v z τ 2 ) 2 ].
z max = z 1 v z τ 2
C st ( X 1 , Z f , X 2 , Z f ,τ )= ( ω 0 ω ) 2 exp( X 1 2 + X 2 2 2 ω 2 )exp{ 2ik v z τ [ X 1 X 2 + 2 M ( v xy τ ) r sp ] 2 1 2 ( v z τ l c /2 ) 2 }
r sp = 2 w 0 M =2 w f
ω= w 0 M 2n1 1+ z 1 2 z F 2
C s ( τ )= C sb ( τ ) C st ( τ )=exp( 2ik v z τ )exp( τ τ b )exp[ ( τ τ t ) 2 ]
τ b = [ ( 2k ) 2 D d ] 1
τ t = [ v x 2 + v y 2 w 0 2 + 1 2 ( v z l c /2 ) 2 ] 1/2 = { v 2 [ ( sin( θ ) w 0 ) 2 + 1 2 ( cos( θ ) l c /2 ) 2 ] } 1/2
c st ( X 1 , Z f , X 2 , Z f ,τ )= C st ( X 1 , Z f , X 2 , Z f ,τ ) I( X 1 , z 1 ) I( X 2 , z 1 ) =exp{ 2ik v z τ [ X 1 X 2 + 2 M ( v xy τ ) r sp ] 2 1 2 ( v z τ l c /2 ) 2 }.
W( f )=2 π τ b U( 2π τ b f, τ b τ t )
U( x,t )=Re[ π 4t e z 2 erfc( z ) ],z= 1ix 2t
L( f )= 1 1 ( 2π τ b ) 2 + f 2 .

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